Problem 49 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 2 \sqrt{4 x+1}=\sqrt{x+4} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Square Both Sides

To eliminate the square roots, we square both sides of the equation:\[(2 \sqrt{4x + 1})^2 = (\sqrt{x + 4})^2\]This simplifies to:\[4(4x + 1) = x + 4\]

Step 2: Expand and Simplify

Distribute the 4 on the left side of the equation:\[16x + 4 = x + 4\]Now, we have a linear equation to solve.

Step 3: Isolate the Variable Term

Subtract \(x\) from both sides to begin isolating \(x\):\[16x - x + 4 = 4\]This simplifies to:\[15x + 4 = 4\]

Step 4: Solve for x

Subtract 4 from both sides:\[15x + 4 - 4 = 4 - 4\]This simplifies to\[15x = 0\]Divide both sides by 15 to find \(x\):\[x = 0\]

Step 5: Check the Solution

Substitute \(x = 0\) back into the original equation to verify:\[2 \sqrt{4(0) + 1} = \sqrt{0 + 4}\]Calculate the expressions inside the square roots:\[2 \cdot \sqrt{1} = \sqrt{4}\]Simplifying, we have:\[2 \cdot 1 = 2\]Which is true, so \(x = 0\) is a valid solution.

Key Concepts

Square Roots

Square roots are symbols used to find a number which, when multiplied by itself, gives the original number. They play a crucial role in solving algebraic equations involving squared terms.

• For an equation like \( \sqrt{a} = b \), it means that when \( b \) is squared, it results in \( a \).
• Square roots help to simplify expressions and solve quadratic equations by "undoing" the squaring of a number.

In the equation \( 2 \sqrt{4x+1} = \sqrt{x+4} \), the square roots need to be eliminated to simplify and solve the equation.
By understanding square roots, you can more confidently tackle algebra problems, knowing how they transform equations.

Squaring Both Sides

Squaring both sides of an equation is a technique to eliminate square roots and make the equation easier to solve.
When you have an equation with square roots, like \( 2 \sqrt{4x + 1} = \sqrt{x + 4} \), squaring both sides helps to remove the radicals:

• This method converts \( (2 \sqrt{4x + 1})^2 \) into \( 4(4x + 1) \) and \( (\sqrt{x + 4})^2 \) into \( x + 4 \).
• The equation becomes \( 4(4x + 1) = x + 4 \), without any square roots.

Squaring both sides is effective, but it requires caution. Sometimes, it can introduce extraneous solutions — solutions that satisfy the squared equation but not the original. Therefore, after squaring, it’s critical to verify any potential solutions.

Isolating the Variable

Isolating the variable in an equation is about getting the variable by itself on one side of the equation, which helps in finding its value.

In practical terms, this involves simplifying the equation by performing operations that will bring all terms with the variable to one side:

• Starting from \( 4(4x + 1) = x + 4 \), distributing the 4 results in \( 16x + 4 \).
• Next, subtract \( x \), then additional constants, from each side to combine like terms and simplify the equation, leading you to \( 15x = 0 \).

Once the variable is isolated, solving becomes straightforward. You simply divide both sides by the numerical coefficient of the variable to find its value.

Checking Solutions

Checking solutions is the final step, ensuring that the calculated solution actually satisfies the original equation. This is crucial, especially with equations that have been squared.

For the equation \( 2 \sqrt{4x+1} = \sqrt{x+4} \):

• After finding \( x = 0 \), you substitute it back into the original equation.
• Calculate the square roots: \( 2 \cdot \sqrt{1} = \sqrt{4} \).
• If the resulting expressions are equal, it confirms that the value is a true solution.

Checking solutions helps identify extraneous solutions, ensuring that only valid answers are considered. Not only does it confirm accuracy, but it also reinforces understanding of the problem-solving process.